Calculus for Business & Economics Notes

Math 140 - Spring 2025

Week 1 Notes

Mon, Jan 13

Today we introduced the course syllabus. Then we looked at how to solve the following equations algebraically.

1. Solve $5-2x = -5$. (https://youtu.be/7GShZYevLGU)

2. Solve $\sqrt{3x + 1} = 4$. (https://youtu.be/0gicD4STzpg)

3. Solve $\dfrac{x}{x-1} = 3$.

After we solved these equations, we also talked about the geometric interpretation of the solutions as places on a graph where a function has a certain y-value.

4. Graph the line $y = 5 - 2x$ and check where it crosses the line $y = -5$.

5. Graph the function $y = \sqrt{3x + 1}$.

Wed, Jan 15

To make it easier to graph functions, it helps to know some basic graphs. Here are six you should memorize.

We used these examples to help graph the following in class:

1. $y = 5|x|$

2. $y = x^2 + 3$ (https://youtu.be/tfF_-Db8iSA?t=2658)

3. $y = \dfrac{1}{x-2}$ (https://youtu.be/tfF_-Db8iSA?t=3391)

4. $y = \sqrt{4 - x}$

After those graphs, we talked about function notation. Both $y = x^2$ and $f(x) = x^2$ to mean the same thing. But the notation $f(x) = x^2$ emphasizes that $f$ is a function of the $x$ variable. Be careful not to confuse function notation $f(5)$ with multiplication $(2)(5) = 10$. Even though the notation looks the same, they are not the same!

We did these examples.

5. If $f(x) = x^2$ and $g(x) = x+5$, find $f(g(4))$ and $g(g(4))$.

6. Find $3 f(2) + 4 g(-1)$.

7. The quantity of gasoline $Q$ sold by a gas station is a function of the price $p$ that the owner sets. Here is a graph of the function $Q = Q(p)$.

   

   1. Use the graph above to find $Q(3)$.
   2. Solve $Q(p) = 3000$ for $p$.
      
   3. If $Q(5) = 600$, explain in English what that means.

8. The function $f(x) = \frac{1}{2}(x + \frac{2}{x})$ can be used to approximate $\sqrt{2}$. Calculate $f(2)$ and $f(f(2))$.

Fri, Jan 17

We started talking about different ways you can combine functions. We did the following exercises.

1. Use the graph below to compute $g(f(-5))$. (https://youtu.be/oORnGaJp1pk)

2. (Exercise 1.2# 29) The function $D(p)$ gives the number of items that will be demanded when the price is $p$. The production cost, $C(x)$ is the cost of producing $x$ items. To determine the cost of production when the price is $6, you would do which of the following?

   1. Evaluate $D(C(6))$
   2. Evaluate $C(D(6))$
   3. Solve $D(C(x)) = 6$
   4. Solve $C(D(p)) = 6$

3. Continuing the previous problem, profit is revenue minus cost, and revenue is price times quantity sold. Using the functions $C$ and $D$, write down formulas for revenue and for profit.

After we talked about function composition, we switched to a quick review of linear functions. You need to know these formulas for linear functions:

• Slope-Intercept Form $y = mx + b$
• Point-Slope Form $y-y_0 = m(x - x_0)$

You also need to understand slope very well:

• Slope $m = \dfrac{\text{rise}}{\text{run}} = \dfrac{\Delta y}{\Delta x} = \dfrac{ y_2 - y_1}{x_2 - x_1}$

4. A line passes through $(-1, 6)$ and $(5, -4)$. Find an equation for the line. (https://youtu.be/XMJ72mtMn4)

Week 2 Notes

Week 3 Notes

Week 4 Notes

Week 5 Notes

Week 6 Notes

Week 7 Notes

Week 8 Notes

Week 9 Notes

Week 10 Notes

Week 11 Notes

Week 12 Notes

Week 13 Notes

Week 14 Notes